Reflection on an semi-transparent object
As I understand it you are trying to illustrate a spherical camera using reflection on a sphere. But your main problem is that the reflection is relative to the position of your (the viewer) perspective.

As you rotate around your sphere, to eventually seeing the backside of it, the reflection of the square gets distorted. For example, in the image below (fig 1) I have positioned a NURBS path to go from the center of the cube to the center of the reflection on the sphere.
Yet as we move around the workspace, we see that the distortion causes the reflection to change its position on the sphere (fig 2):


If you wish for the "reflections" (if we can even call it that at this point) to be mapped to the sphere regardless of camera position, then you need baking. (animated gif):



How I accomplished this:


First I set up my scene: added a background, the sphere, some props from the archimesh addon and a cube with black lines around it.
I selected the sphere and went into edit mode, turned on orthographic mode, and snapped to front view via numpad-1.
I unwrapped the sphere using "sphere projection" (u key) and opened the node editor and the uv/image editor.
I created a new "blank image" in the image viewer, and set up the material for the sphere, like so: Note how the Image node is not connected. It is also important that the image node uses the new blank image, and that the image node is the selected node.
With the sphere selected I go to the render panel, under "bake", select "glossy" and click the bake button:

As you can see in the image editor, the "blank new image" is no longer blank:

Then I reconfigured the sphere material to use the baked image, taking into account the transparency you asked for, etc


After this, it works! However, there are some caveats:


Because it's a baked image, it doesn't work well when the environment around it is changing. To get by this you would have to bake an image sequence of it per frame or take advantage of the equirectangular uv mapping and somehow pipe a camera into its texture (not sure if that's even possible).
It's annoying to have to rebake every time you change something.


Good luck and happy blending!